Complete gradient shrinking Ricci solitons have finite topological type

Fu-quan Fang; Jian-wen Man; Zhen-lei Zhang

doi:10.1016/j.crma.2008.03.021

Differential Geometry

Complete gradient shrinking Ricci solitons have finite topological type

Presented by: Étienne Ghys

Fu-quan Fang ¹; Jian-wen Man ²; Zhen-lei Zhang ²

¹ Department of Mathematics, Capital Normal University, Beijing, PR China
² Chern Institute of Mathematics, Weijin Road 94, Tianjin 300071, PR China

Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 653-656.

Abstract
Résumé

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:

(i) the Ricci curvature is bounded from above;

(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.

Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.

Dans cette Note nous montrons qu'une variété riemanienne complète est de type topologique fini – c'est-à-dire qu'elle est homéomorphe à une variété compacte à bord – si son tenseur de Bakry–Emery–Ricci est bornée inférieurement par une constante positive et vérifie l'une des conditions suivantes :

(i) la courbure de Ricci est bornée supérieurement.

(ii) la courbure de Ricci est bornée inférieurement et le rayon d'injectivité est positif.

De plus, un soliton de Ricci contractant complet est de type topologique fini si sa coubure scalaire est bornée.

Received: 2008-01-03
Accepted: 2008-03-18
Published online: 2008-04-29

DOI: 10.1016/j.crma.2008.03.021

Author's affiliations:

Fu-quan Fang ¹; Jian-wen Man ²; Zhen-lei Zhang ²

¹ Department of Mathematics, Capital Normal University, Beijing, PR China
² Chern Institute of Mathematics, Weijin Road 94, Tianjin 300071, PR China

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     author = {Fu-quan Fang and Jian-wen Man and Zhen-lei Zhang},
     title = {Complete gradient shrinking {Ricci} solitons have finite topological type},
     journal = {Comptes Rendus. Math\'ematique},
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     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.03.021},
     language = {en},
}

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Fu-quan Fang; Jian-wen Man; Zhen-lei Zhang. Complete gradient shrinking Ricci solitons have finite topological type. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 653-656. doi : 10.1016/j.crma.2008.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.021/

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